Penalty Robin-Robin Domain Decomposition Schemes 2 for Contact Problems of Nonlinear Elastic Bodies 3 UNCORRECTED PROOF

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1 1 Penalty Robin-Robin Domain Decomposition Schemes 2 for Contact Problems of Nonlinear Elastic Bodies 3 Ihor I. Prokopyshyn 1, Ivan I. Dyyak 2, Rostyslav M. Martynyak 1,and 4 Ivan A. Prokopyshyn IAPMM NASU, Naukova 3-b, Lviv, 79060, Ukraine, ihor84@gmail.com 6 2 Ivan Franko National University of Lviv, Universytetska 1, Lviv, 79000, Ukraine 7 1 Introduction 8 Many domain decomposition techniques for contact problems have been proposed 9 on discrete level, particularly substructuring and FETI methods [1, 4]. 10 Domain decomposition methods (DDMs), presented in [2, 10, 11, 16] for unilat- 11 eral two-body contact problems of linear elasticity, are obtained on continuous level. 12 All of them require the solution of nonlinear one-sided contact problems for one or 13 both of the bodies in each iteration. 14 In works [6, 14, 15] we have proposed a class of penalty parallel Robin Robin 15 domain decomposition schemes for unilateral multibody contact problems of linear 16 elasticity, which are based on penalty method and iterative methods for nonlinear 17 variational equations. In each iteration of these schemes we have to solve in a parallel 18 way some linear variational equations in subdomains. 19 In this contribution we generalize domain decomposition schemes, proposed in 20 [6, 14, 15] to the solution of unilateral and ideal contact problems of nonlinear elastic 21 bodies. We also present theorems about the convergence of these schemes Formulation of Multibody Contact Problem 23 Consider a contact problem of N nonlinear elastic bodies Ω α R 3 with sectionally 24 smooth boundaries Γ α, α = 1,2,...,N (Fig. 1). Denote Ω = N α=1 Ω α. 25 A stress-strain state in point x =(x 1,x 2,x 3 ) of each body Ω α is defined by the 26 displacement vector u α = u α i e i, the tensor of strains ˆε α = ε α ij e i e j and the tensor 27 of stresses ˆσ α = σ α ij e i e j. These quantities satisfy Cauchy relations, equilibrium 28 equations and nonlinear stress-strain law [8]: 29 σ α ij = λ α δ ij Θ α + 2 μ α ε α ij 2 μ α ω α (e α )e α ij, i, j = 1,2,3, (1) where Θ α = ε α 11 + ε α 22 + ε α 33 is the volume strain, λ α (x) > 0, μ α (x) > 0are 30 bounded Lame parameters, e α ij = ε α ij δ ij Θ α / 3 are the components of the strain 31 R. Bank et al. (eds.), Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering 91, DOI / , Springer-Verlag Berlin Heidelberg 2013 Page 679

2 Ihor I. Prokopyshyn Fig. 1. Contact of several bodies deviation tensor, e α = 2g α / 3 is the deformation intensity, gα =(ε α11 ε α22 ) (ε α22 ε α33 ) 2 +(ε α33 ε α11 ) 2 + 6(ε 2 α12 + ε2 α23 + ε2 α31 ), and ω α(z) is nonlinear dif- 33 ferentiable function, which satisfies the following properties: 34 0 ω α (z) (zω α (z)) / z < 1, (ω α (z)) / z 0. (2) On the boundary Γ α let us introduce the local orthonormal basis ξ α, η α, n α, 35 where n α is the outer unit normal to Γ α. Then the vectors of displacements and 36 stresses on the boundary can be written in the following way: u α = u αξ ξ α + 37 u αη η α + u αn n α, σ α = ˆσ α n α = σ αξ ξ α + σ αη η α + σ αn n α. 38 Suppose that the boundary Γ α of each body consists of four disjoint parts: Γ α = 39 Γα u Γ σ α Γ I α Sα, Γα u /0, Γ α u = Γ α u, Γ α I Sα /0, where S α = β B α S αβ,and 40 Γα I = β I α Γ αβ. Surface S αβ is the possible unilateral contact area of body Ω α with 41 body Ω β,andb α {1,2,...,N} is the set of the indices of all bodies in unilateral 42 contact with body Ω α. Surface Γ αβ = Γ β α is the ideal contact area between bodies 43 Ω α and Ω β,andi α {1,2,...,N} is the set of the indices of all bodies which have 44 ideal contact with Ω α. 45 We assume that the areas S αβ Γ α and S βα Γ β are sufficiently close (S αβ 46 S βα ), and n α (x) n β (x ), x S αβ, x = P(x) S βα,wherep(x) is the projection 47 of x on S αβ [12]. Let d αβ (x) =± x x 2 be a distance between bodies Ω α and 48 Ω β before the deformation. The sign of d αβ depends on a statement of the problem. 49 We consider homogenous Dirichlet boundary conditions on the part Γα u, and Neu- 50 mann boundary conditions on the part Γα σ : 51 u α (x)=0, x Γ u α ; σ α (x)=p α (x), x Γ σ α. (3) On the possible contact areas S αβ, β B α, α = 1,2,...,N the following nonlin- 52 ear unilateral contact conditions hold: 53 σ αn (x)=σ β n (x ) 0, σ αξ (x)=σ βξ (x )=σ αη (x)=σ βη (x )=0, (4) u αn (x)+u β n (x ) d αβ (x), (5) 54 Page 680

3 Penalty Robin-Robin Domain Decomposition Schemes ( uαn (x)+u β n (x ) d αβ (x) ) σ αn (x)=0, x S αβ, x = P(x) S βα. (6) 55 On ideal contact areas Γ αβ = Γ β α, β I α, α = 1,2,...,N we consider ideal 56 mechanical contact conditions: 57 u α (x)=u β (x), σ α (x)= σ β (x), x Γ αβ. (7) 3 Penalty Variational Formulation of the Problem 58 For each body Ω α consider Sobolev space V α =[H 1 (Ω α )] 3 and the closed subspace 59 Vα 0 = {u α V α : u α = 0onΓα u}. All values of the elements u α V α, u α Vα 0 on 60 the parts of boundary Γ α should be understood as traces [9]. 61 Define Hilbert space V 0 = V V N 0 with the scalar product (u,v) V 0 = 62 N α=1 (u α,v α ) Vα and norm u V0 = (u,u) V0, u,v V 0. Introduce the closed con- 63 vex set of all displacements in V 0, which satisfy nonpenentration contact condi- 64 tions (5) and ideal kinematic contact conditions: 65 K = { u V 0 : u α n + u β n d αβ on S αβ, u α = u β on Γ α β }, (8) where {α, β } Q, Q = {{α,β } : α {1,2,...,N}, β B α }, {α, β } Q I, Q I = 66 {{α,β } : α {1,2,...,N}, β I α },andd αβ H 1/2 00 (Ξ α), Ξ α = int(γ α \ Γα u ). 67 Let us introduce bilinear form A(u,v)= N α=1 a α(u α,v α ), u,v V 0, which rep- 68 resents the total elastic deformation energy of the system of bodies, linear form 69 L(v) = N α=1 l α(v α ), v V 0, which is equal to the external forces work, and non- 70 quadratic functional H (v)= N α=1 h α(v α ), v V 0, which represents the total nonlin- 71 ear deformation energy: 72 a α (u α,v α )= [λ α Θ α (u α )Θ α (v α )+2μ α Ω α ε α ij (u α )ε α ij (v α )]dω, (9) i, j l α (v α )= f α v α dω + p α v α ds, (10) Ω α Γα σ eα (v α ) h α (v α )=3 zω α (z) dz dω, (11) Ω α μ α 0 where p α [H 1/2 00 (Ξ α )] 3,andf α [L 2 (Ω α )] 3 is the vector of volume forces. 75 Using [12], we have shown that the original contact problem has an alternative 76 weak formulation as the following minimization problem on the set K: F(u)=A(u,u)/2 H(u) L(u) min u K. (12) Bilinear form A is symmetric, continuous with constant M A > 0 and coercive 78 with constant B A > 0, and linear form L is continuous. Nonquadratic functional H is 79 doubly Gateaux differentiable in V 0 : 80 Page 681

4 Ihor I. Prokopyshyn H (u,v)= α h α(u α,v α ), H (u,v,w)= h α(u α,v α,w α ), u,v,w V 0, (13) h α (u α,v α )=2 α μ α ω α (e α (u α )) Ω α e αij (u α )e αij (v α ) dω. (14) i, j Moreover, we have proved that the following conditions hold: 82 ( C > 0)( u V 0 ) {(1 C)A(u,u) 2H (u) }, (15) { H 83 ( u V 0 )( R > 0)( v V 0 ) (u,v) R v V0}, (16) { H 84 ( D > 0)( u,v,w V 0 ) (u,v,w) D v V0 w V0}, (17) } 85 ( B > 0)( u,v V 0 ) {A(v,v) H (u,v,v) B v 2 V0. (18) From these properties, it follows that there exists a unique solution ū K of min- 86 imization problem (12), and this problem is equivalent to the following variational 87 inequality, which is nonlinear in u: 88 A(u,v u) H (u,v u) L(v u) 0, v K, u K. (19) To obtain a minimization problem in the whole space V 0, we apply a penalty 89 method [3, 7, 9, 13] to problem (12). We use a penalty in the form 90 J θ (u)= 1 2θ ( ) 2 d αβ u α n u β n + L 2 (S αβ ) {α, β } Q + 1 2θ u α u β 2, [L 2 (Γ {α, β } Q I α β )] 3 (20) where θ > 0 is a penalty parameter, and y = min{0,y}. 91 Now, consider the following unconstrained minimization problem in V 0 : 92 F θ (u)=a(u,u)/2 H(u) L(u)+J θ (u) min. (21) u V 0 The penalty term J θ is nonnegative and Gateaux differentiable in V 0, and its dif- 93 ferential J θ (u,v)= 1 θ ( ) ( ) {α, β } Q S dαβ αβ u α n u β n vα n + v β n ds+ 94 ( ) ( ) 1 uα u β vα v β ds satisfy the following properties [15]: 95 θ {α, β } Q I Γ α β ( u V 0 )( R > 0)( v V 0 ) { J θ (u,v) R v V0}, (22) { J ( D > 0)( u,v,w V 0 ) θ (u + w,v) J θ (u,v) D v w V0 V0}, (23) ( u,v V 0 ) { J θ (u + v,v) J θ (u,v) 0}. (24) Using these properties and the results in [3], we have shown that problem (21) 98 has a unique solution ū θ V 0 and is equivalent to the following nonlinear variational 99 equation in the space V 0 : 100 F θ (u,v)=a(u,v) H (u,v)+j θ (u,v) L(v)=0, v V 0, u V 0. (25) Using the results of works [7, 13], we have proved that ū θ ū V0 θ Page 682

5 Penalty Robin-Robin Domain Decomposition Schemes 4 Iterative Methods for Nonlinear Variational Equations 102 In arbitrary reflexive Banach space V 0 consider an abstract nonlinear variational 103 equation 104 Φ (u,v)=l(v), v V 0, u V 0 (26) where Φ : V 0 V 0 R is a functional, which is linear in v, but nonlinear in u, 105 and L is linear continuous form. Suppose that this variational equation has a unique 106 solution ū V For the numerical solution of (26) we use the next iterative method [5, 6, 15]: 108 [ ] G(u k+1,v)=g(u k,v) γ Φ (u k,v) L(v), v V 0, k = 0,1,..., (27) where G is some given bilinear form in V 0 V 0, γ R is fixed parameter, and u k V is the k-th approximation to the exact solution of problem (26). 110 We have proved the next theorem [5, 15] about the convergence of this method. 111 Theorem 1. Suppose that the following conditions hold 112 } ( u V 0 )( R Φ > 0)( v V 0 ) { Φ (u,v) R Φ v V0, (28) } ( D Φ >0)( u,v,w V 0 ) { Φ (u + w,v) Φ (u,v) D Φ v V0 w V0, (29) } ( B Φ > 0)( u,v V 0 ) {Φ (u + v,v) Φ (u,v) B Φ v 2 V0, (30) bilinear form G is symmetric, continuous with constant M G > 0 and coercive with 115 constant B G > 0, and γ (0; 2γ ),γ = B Φ B G /D 2 Φ. 116 Then u k ū V0 0,where{u k } V 0 is obtained by method (27). Moreover, 117 k the convergence rate in norm G = G(, ) is linear, and the highest convergence 118 rate in this norm reaches as γ = γ. 119 In addition, we have proposed nonstationary iterative method to solve (26), where 120 bilinear form G and parameter γ are different in each iteration: 121 [ ] G k (u k+1,v)=g k (u k,v) γ k Φ (u k,v) L(v), v V 0, k = 0,1,.... (31) A convergence theorem for this method is proved in [15] Domain Decomposition Schemes for Contact Problems 123 Now let us apply iterative methods (27) and(31) to the solution of nonlinear penalty 124 variational equation (25) of multibody contact problem. This penalty equation can 125 be written in form (26), where 126 Φ (u,v)=a(u,v) H (u,v)+j θ (u,v), u,v V 0. (32) Page 683

6 Ihor I. Prokopyshyn We consider such variants of methods (27) and(31), which lead to the domain 127 decomposition. 128 Let us take the bilinear form G in iterative method (27) as follows [6, 15]: 129 X(u,v)= 1 θ N α=1 G(u,v)=A(u,v)+X(u,v), u,v V 0, (33) [ ] u α n v α n ψ αβ ds+ u α v α φ αβ ds S αβ β B α β I α Γ αβ 130, 131 where ψ αβ (x)={1, x S 1 αβ } {0, x S αβ\s 1 αβ } and φ αβ (x)={1, x Γ αβ 1 } 132 {0, x Γ αβ \Γαβ 1 } are characteristic functions of arbitrary subsets S 1 αβ S αβ, 133 Γαβ 1 Γ αβ of possible unilateral and ideal contact areas respectively. 134 Introduce a notation ũ k+1 =[u k+1 u k ]/γ + u k V 0. Then iterative method (27) 135 with bilinear form (33) can be written in such way: 136 ( ) ( ) ( ) A ũ k+1,v + X ũ k+1,v = L(v)+X u k,v + H (u k,v) J θ (uk,v), (34) u k+1 = γ ũ k+1 +(1 γ)u k, k = 0,1,.... (35) Bilinear form X is symmetric, continuous with constant M X > 0, and nonnegative 138 [15]. Due to these properties, and due to the properties of bilinear form A, it follows 139 that the conditions of Theorem 1 hold. Therefore, we obtain the next proposition: 140 Theorem 2. The sequence {u k } of the method (34) (35) converges strongly to the 141 solution of penalty variational equation (25) forγ (0; 2B Φ B G /D 2 Φ ),whereb G = 142 B A,B Φ = B, D Φ = M A + D + D. The convergence rate in norm G is linear. 143 As the common quantities of the subdomains are known from the previous iter- 144 ation, variational equation (34) splits into N separate equations for each subdomain 145 Ω α, and method (34) (35) can be written in the following equivalent form: 146 a α (ũ k+1 α,v ψ αβ α)+ ũ k+1 α n β B S α αβ θ v φ αβ α n ds+ ũ k+1 α v α ds 147 β I Γ α αβ θ 148 = l α (v α )+ 1 θ ( ) ] [ψ αβ u kα n + d αβ u k α n uk β n v α n ds 149 β B S α αβ + 1 θ [ φ αβ u k α + β I Γ α αβ ( )] u k β uk α v α ds+ h α(u k α,v α ), v α Vα, 0 (36) u k+1 α = γ ũ k+1 α +(1 γ)u k α, α = 1,2,...,N, k = 0,1,.... (37) In each iteration k of method (36) (37), we have to solve N linear variational 152 equations in parallel, which correspond to some linear elasticity problems in sub- 153 domains with additional volume forces in Ω α, and with Robin boundary conditions 154 on contact areas. Therefore, this method refers to parallel Robin Robin type domain 155 decomposition schemes. 156 Page 684

7 Penalty Robin-Robin Domain Decomposition Schemes Taking different characteristic functions ψ αβ and φ α β, we can obtain different 157 particular cases of penalty domain decomposition method (36) (37). 158 Thus, taking ψ αβ (x) 0, β B α, φ αβ (x) 0, β I α, α = 1,2,...,N, weget 159 parallel Neumann Neumann domain decomposition scheme. 160 Other borderline case is when ψ αβ (x) 1, β B α, φ αβ (x) 1, β I α, α = 161 1,2,...,N, i.e.s 1 αβ = S αβ, Γαβ 1 = Γ αβ. 162 Moreover, we can choose functions ψ αβ and φ αβ differently in each iteration k. 163 Then we obtain nonstationary domain decomposition schemes, which are equivalent 164 to iterative method (31) with bilinear forms 165 G k (u,v)=a(u,v)+x k (u,v), u,v V 0, k = 0,1,..., (38) [ 166 X k (u,v)= 1 N θ u α n v α n ψ k αβ ds+ u α v α φ k αβ ]. S ds 167 αβ α=1 β B α β I α If we take characteristic functions ψ k αβ and φ k αβ as follows [6, 14, 15]: 168 ψ k αβ (x)=χk αβ (x)= { 0, dαβ (x) u k α n(x) u k β n (x ) 0 1, d αβ (x) u k α n (x) uk β n (x ) < 0, x = P(x), x S αβ, 169 φ k αβ (x) 1, x Γ αβ, β B α, β I α, α = 1,2,...,N, 171 then we shall get the method, which can be conventionally named as nonstationary 172 parallel Dirichlet Dirichlet domain decomposition scheme. 173 In addition to methods (27), (33)and(31), (38), we have proposed another family 174 of DDMs for the solution of (25), where the second derivative of functional H(u) is 175 used. These domain decomposition methods are obtained from (31), if we choose 176 bilinear forms G k (u,v) as follows 177 G k (u,v)=a(u,v) H (u k,u,v)+x k (u,v), u,v V 0, k = 0,1,.... (39) Numerical analysis of presented penalty Robin Robin DDMs has been made 178 for plane unilateral two-body and three-body contact problems of linear elasticity 179 (ω α 0) using finite element approximations [6, 14, 15]. Numerical experiments 180 have confirmed the theoretical results about the convergence of these methods. 181 Among the positive features of proposed domain decomposition schemes are 182 the simplicity of the algorithms and the regularization of original contact problem 183 because of the use of penalty method. These domain decomposition schemes have 184 only one iteration loop, which deals with domain decomposition, nonlinearity of the 185 stress-strain relationship, and nonlinearity of unilateral contact conditions. 186 Γ αβ 170 Bibliography 187 [1] P. Avery and C. Farhat. The FETI family of domain decomposition meth- 188 ods for inequality-constrained quadratic programming: Application to contact 189 problems with conforming and nonconforming interfaces. Comput. Methods 190 Appl. Mech. Engrg., 198: , Page 685

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